The Optimal Management of a Natural Resource with

Switching Dynamics

Michele Baggio

Chair of Environmental Policy and Economics, ETH ZurichUniversitaetstr. 22, CHN 76.2, 8092 Zurich, Switzerland

michele.baggio@env.ethz.ch

Abstract

This paper analyzes the optimal management of a natural resource, such asa fishery, where the dynamics of the resource shift between different statesat random times due to the effect of external forcing such as climate. Thismeans that the parameters describing the biological relationships are differ-ent under different states. Using a classical linear control model I investigatehow the switching behavior influences the optimal exploitation of the re-source. The optimal switching conditions are used to identify the thresholdsdetermining the status, opening/closing, of the fishing industry. The modelis applied to the Peruvian anchoveta fishery located along the north-centralcoast of Peru. The optimal management policy that is constituted by fourthresholds defining the stock levels at which the industry switches status un-der each state of the stock dynamics. Further, I show how such thresholds areinfluenced by the probability of a regime shift and other key parameters ofthe model, e.g., the maximum capacity of the fishing industry. This analysisgives important indications for the management of a natural resource withalternating dynamics, which can be used to design policies that adapt to thevariability of the physical environment.

Keywords: Bioeconomic modeling, Fisheries, Peruvian anchoveta, Realoption theory, Regime shiftsJEL: C61, Q22, Q57

Preliminary draft. Please do not cite May 14, 2012

1. Introduction

It is typical for marine ecosystems to fluctuate around some persistent trend

or equilibrium. However, marine systems are occasionally exposed to trans-

formations by means of either natural phenomena or anthropogenic interven-

tions (or both) which may cause sudden shifts in the state of ecosystems (e.g.,

Scheffer et al., 2001). In the ecological literature, this is referred to as regime-

switching (Lees et al., 2006). A regime shift can induce drastic drops in the

abundance of a population or changes in the biological/ecological structure of

communities, like changes in the growth rate. Examples include the shifting

between oligotrophic and eutrophic states in lakes (Carpenter et al., 1999),

the oscillation of the dominant species in oceans, e.g., anchovies and sardines

in the Pacific (Chavez et al., 2003), or the collapse of fish stocks like the case

of the collapse of the Georges Bank Haddock fishery (1960s-1990s) (Collie

et al., 2004).

When the dynamics of a natural resource can switch between alternative

stable equilibrium and once the threshold between these equilibria is crossed

it may be difficult to push the system back to the original equilibrium (Car-

penter et al., 1999; Scheffer et al., 2001; Peterson et al., 2003). This has

implications on the management of the natural resource. In fact, the switch-

ing between equilibria could induces a hysteretic behavior of the optimal

management policy (Carpenter et al., 1999; Scheffer et al., 2001).

Scientists advocate that the management of fisheries should incorporate

regime switching and gain a better understanding of their determinants (e.g.,

Beamish et al., 2004; Collie et al., 2004; Rothschild and Shannon, 2004).

Some (e.g., Polovina, 2005; King and McFarlane, 2006) have suggested the

2

use of regime-specific harvest rates, such as with low harvest rate, or even

closing the fishery, during low productivity regimes and a high rate in the

high productivity regimes. Therefore, it is necessary to identify the nature of

the shifts to design management strategies that adapt or mitigate the effects

of the shifts (deYoung et al., 2008).

Bakun (2005, p.974) defines as regime shifts in marine ecosystems a “per-

sistent radical shift in typical levels of abundance or productivity of multiple

important components of marine biological community structure, occurring

at multiple trophic levels and on a geographical scale that is at least regional

in extent.” This paper analyzes the optimal management of a fishery in

which the dynamics of the fish stock shift between different states at random

times. Like in (Hamilton, 1989), regime shifts in a time series are defined

as “episodes across which the behavior of the series is markedly different.”

Considering the dynamics of a natural resource stock, this implies that the

dynamics of the stock is different under the different regimes, i.e., the pa-

rameters describing the biological relationships are different under different

regimes.

Using the classical linear control by Clark and Munro (1975) I investigate

how regime switching stock dynamics influence the optimal extraction of the

resource. I assume that any variation in the harvesting rate involves a lump-

sum payment. The presence of switching costs implies that the agent may

delay opening (closing) to harvest if the value of the resource is not large

enough to sustaining the opening (closing) costs (e.g., Nøstbakken, 2006).

When the stock switches between regimes, the harvest strategy is less clear.

An important question is whether expectations toward regime shifts actually

3

affect the way an agency manages the fishery. Since managerial decisions

over the status of the resource varies in response to the state followed by the

natural resource, the number of critical points on which the harvest strategy

is based increases. Hence, the optimal policy involves opening/closing the

resource to exploitation when the thresholds are crossed.

Despite the vast treatment of regime switching in the ecological litera-

ture, the integration of the role of regime switching behavior in the fisheries

economic literature remains incomplete. Recently, (Polasky et al., 2011) de-

veloped an optimal management resource model that incorporates both a

potential collapse and switching dynamics of a natural resource. Allowing

for both exogenous and endogenous shits, they assume that sometime in a

future period the resource will shift to an alternative regime from which it

will never return. Unlike Polasky et al. (2011), in this paper the dynamic

of the resource is allowed to switch back and forth between two alternative

regimes at any point in time.1

I derive solutions of model that are used to identify the critical values,

thresholds, that determine the opening/closing of the fishing industry. Using

a numerical analysis, I show how such thresholds are influenced by the prob-

ability the stock will switch regime and other key parameters of the model,

e.g., the drift and the variance of the stochastic process describing the natural

stock. Results indicate that the thresholds defining the optimal management

strategy are highly sensitive to changes in the model’s parameters.

1The shift is first treated as exogenous, but the model will be extended to endogenousswitching in its further development. In fact, the next step of this analysis involves anextension of this model which will allow switching probabilities to be endogenous, that isto depend on the stock itself and some exogenous (natural) conditions.

4

The article is organized as follows. The next section introduces the theo-

retical model of the optimal management of a natural resource with regime

switching behavior. The optimal solutions are derived in section three. The

illustration using data on the Peruvian anchoveta fishery, is presented in sec-

tion four. The results of the numerical analysis are then presented in section

five. Finally, [...]

2. The Model

Consider an exploited natural resource like a fish stock, which is denoted

by x. Let assume that the transition process describing the dynamics of

the resource can shift between two alternative states, which differ in the

characteristics of the biological structure of the resource. In other words,

the parameters defining the dynamics of the resource are different under the

two states. This behavior is described by a Markov regime switching model

where the parameters of the stock dynamics shift between states

dxt = µs(t)(xt, yt)dt+ σs(t)(xt)dWt, ∀xt > 0, (1)

where{s(t)}

is a Markov process denoting the state of the resource, y denotes

the resource that is consumed, e.g., harvest in case of a fishery, and dW is

an increment of a Wiener process independent of s with dW = ut√dt, ut ∼

N(0, 1), such that E(utut′) = 0 for t 6= t′, and E(dW ) = 0, E(dtdW ) =

0, Var(dW ) = dt (Kamien and Schwartz, 1991), where E symbolizes the

expectation operator and Var indicates the variance.

The two components of the process, µs(x, y) and σs(x) are known func-

tions that correspond to the expected drift rate and the volatility of the

5

resource. The drift rate represents the resource growth function, which is

assumed to be linear in y. The growth function and the volatility can take

different values when the process s is in different state. Having a different

drift term for each regime implies having different biological parameters of

the population under each state. For instance, the nature of the biological

relationship may be qualitatively the same, but the magnitude of growth rate

or natural mortality is different under different states.

As in Guo et al. (2005), it is assumed that s is observable and that

the probabilities of moving between states follows a Poisson law such that

this process is a two-state Markov chain alternating between two states, i.e.,

s = 1, 2.2 Let λsdt denote the (transition) probability of leaving state s. In

other words this is the probability to shift from state s to the alternative state

s′ in the interval time dt. Thus, a complete representation of the transition

between states between time t and t+ dt is given by the probability matrix

Λ =

1− λ1dt λ1dt

λ2dt 1− λ2dt

.The Markov chain is assumed to be irreducible such that there are no ab-

sorbing states, i.e., 0 < λsdt < 1, i = 1, 2, meaning that there exists no

irreversible state of the stock dynamics.3 This implies that in each time pe-

riod there is a positive probability that drift and volatility may change as a

results of the regime shift.

2Perrings (1998) and Rothschild and Shannon (2004) suggest the use of Markov chainsto model the transition between alternative regimes, or states, of an ecosystem.

31− λsdt could be interpreted as the resilience of state s.

6

The resource is exploited by an industry that extracts rents for the soci-

ety. An economic agent such as a social planner or agency that chooses the

harvest policy that maximizes the present value of the future stream of rents

extracted from the industry subject to the dynamics of the resource (1). The

objective function of the problem is:

V (x, s) = max0≤y≤ymax

E

{∫ ∞0

e−δt (p− c(x)) y dt

}, (2)

where V (x, s) denotes the current value function, c(x) is the harvesting cost,

ymax indicates the maximum feasible harvest, and δ is the discount factor.

Because the stock dynamics depend on the current regime, at each instant

the relevant state variables are {(x, s) : x ∈ <+, s = 1, 2}. This implies that

also the value-maximizing harvesting policy depends on the state as well as

the resource.

In the deterministic case, when µ1 = µ2 and σ1 = σ2 = 0, because the

problem is linear in the control variable the optimal harvest policy follows a

bang-bang approach with a constant harvest rate (Clark and Munro, 1975).

When the stock is above its steady-state level, it is optimal to decrease the

stock as rapidly as possible with y = ymax. Notice that it is assumed that

ymax is large enough to drive the stock down. When the stock is below the

steady-state level, it is optimal not to harvest, y = 0, so that the stock can

be quickly rebuilt. Hence, at every point in time, the optimal harvest pol-

icy follows the maximum rapid approach path (MRAP) toward the optimal

(steady-state) level (Spence and Starrett, 1975).

The optimal harvest policy involves either fishing at maximum capacity

or not fishing at all. Thus, changing harvest rate involves either opening

7

or closing the fishing industry which requires a lump-sum payment. For

instance, opening the fishery entails bearing administrative costs such as for

setting up enforcement/monitoring system, while closing the fishery could

imply compensating the fishermen left idle. Let Mo and Mc denote opening

and closing costs, respectively. The problem is still linear in the control

variable and the MRAP solution is better than harvesting at a constant rate

at all times (Brekke and Øksendal, 1994; Nøstbakken, 2006).

In the presence of start-up or shut-down costs the agent may delay open-

ing to harvest if the profitability of the fishery is not large enough to sus-

taining the opening costs. Similarly, with positive closing costs closing the

fishery may be delayed even if the stock level is lower than the steady-state

level, which would induce the closure in the zero-fixed cost case. Therefore,

the critical stock level that would induce a closed fishery to open exceeds the

stock level that would induce an open fishery to close. Hence, similarly to the

well known case of entry and exit decisions of a firm under uncertainty (e.g.,

Dixit, 1989), the status of the fishery (open/closed) is determined according

to not just one but two critical levels of the stock.

In the model presented here, though, the dynamic of the resource stock

may switch between states. This introduces another source of uncertainty

that creates a richer set of strategies than the deterministic model or even

for the stochastic single-regime stock with fixed costs. The presence of fixed

costs and a switching dynamics creates a “double type” of hysteresis, with

two critical values for each state of the resource dynamics.4 It is worth noting

4In the single state case, hysteresis indicates the gap between opening and closingthresholds. In the two states case, it indicates the gap between opening, or closing, thresh-

8

that even in the absence of fixed costs the agent could delay or anticipate

the change in the status of the industry due to the expectation of a shift in

the state of the resource. For instance, given that the resource is in a less

favorable state, the agent may delay stopping harvest if he anticipates that

the resource will shift to a more favorable state.

To clarify, let x∗s denote the level that triggers the opening of the fishery

under state s. In each state, for stock values above this threshold the value

of an open fishery weakly exceeds the value of a closed fishery plus the cost

incurred by opening the industry to harvest; and the optimal harvest policy

is to fish at maximum capacity. On the contrary, x∗∗s denotes the stock

level that triggers the closing of the fishery under state s, that is, below this

threshold the fishery is closed. This is because the value of a closed fishery

weakly exceeds the value of an open fishery plus the closing costs. In this

case no harvesting occurs. In the range between the two critical points the

fishery remains in its current status. If the fishery is closed, the fishery will

remain closed because the agent holds an option to open, but even though

opening would generate profit, this would be small enough that the agent may

delay opening until conditions improve, that is when the upper threshold x∗s

is crossed (from below). The other case arises when an open fishery will

remain open and harvest will be set at its maximum level until the lower

threshold x∗∗s will be crossed (from above). At that point it is optimal to

shut down the industry.

The cases just described arise when each regime is considered separately.

But how different is the situation if at any point in time the dynamic of

olds for the two states.

9

the stock can switch regime and changing harvest rate is costly? There are

instances where the status of the fishery (open/closed) is the same as in the

single regime case. But, there are also values of stock for which different

behavior can arise. This is addressed next.

3. Solving the Regime Switching Problem

It is assumed that the two states can differ due to different growth rate of

the resource so that the resource can be either in a high productive (H) or

low productive state (L). In this case, there exist four regimes, indexed by R,

which are {high, closed} and {high, open} when the resource is highly pro-

ductive and the harvest rate equals zero or ymax, respectively, {low, closed}

and {low, open} when the resource is less productive and the harvest rate

equals zero or ymax, respectively, with R = 1, . . . , 4.

The solution of the problem (1–2) is given by the stock thresholds defining

the range of stocks for four regimes that are given by the combination of

the states of the stock dynamics and two status for the fishing industry,

open/closed. Thus, the solution consists of a pair of critical values for each

state of the natural resource leading to existence of four regimes.

Under either state of the resource dynamics, high or low, the thresholds

triggering the opening are denoted as (x∗H , x∗L), and those triggering the clos-

ing are (x∗∗H , x∗∗L ). The order of the trigger points depends on the parameters

of the model, mainly the switching costs and the transition probabilities. Un-

der the assumption that the resource is more productive under state H than

L, let consider the configuration where the thresholds follow the sequence:

10

x∗∗H < x∗∗L < x∗H < x∗L (figure 3).5 For values below x∗∗H , regardless the state,

it is optimal to keep the fishery closed at all time. The opposite is true for

values above x∗L. In that case it is optimal to keep the fishery open. For the

other ranges of stock value the optimal behavior is less clear. In the range

x∗∗H < x < x∗∗L , under state H an open (closed) fishery would remain open

(closed). But what happens in case the stock shifts to state L? Following a

shift to state L, it would be optimal to shut down the fishery because the

value of a closed fishery (the value or the option to enter) exceeds the value

of an open fishery plus the closing costs. For this range, in state L the fishery

is closed at all time and a shift in the stock dynamics would not change the

status of the fishery.

The range x∗H < x < x∗L presents a similar situation although for reversed

states. Under state L, a closed fishery would remain closed unless the stock

shifts to state H, which would induce the fishery to open. When it is open,

the fishery remains open no matter what is the state. Notice that under state

H it is never optimal to keep the fishery closed. Finally, when the stock is in

the range x∗L < x < K the fishery is always open no matter the state of the

stock dynamic.

Solutions of optimal switching problems are provided by (e.g.) Brekke

and Øksendal (1994), Fackler and Sengupta (2005), and Guo et al. (2005)

Following Fackler and Sengupta (2005), the HJB equation associated with

5For example it could be possible that the maximum attainable population K is lessthan one of the thresholds; meaning that for high opening costs it would be best to waituntil the stock reaches a very high level.

11

the optimization problem is the following equation:

δ V (x,R) = max0≤y≤ymax

{(p− c(x)) y + µR(x, y)Vx

+1

2σ2r(x)Vxx +

∑j

λR,j [V (x, j)− V (x,R)−MR,j]

} (3)

where λR,j is the instantaneous probability of switching from the current

regime R to regime j; notice that 1−∑

j λR,j is the probability of remaining in

regime R. As before, each change in the state requires a lump-sum payment

(fixed cost) of MR,j. The term (p − c(x)) y is the immediate profit, and

the remainder of the right-hand side is the continuation value (the outcome

of future optimal decisions). The last term corresponds to the expected

change in the value of the fishery due to a regime shift given by the value

of the resource after the switch minus the value of the resource before the

switch and the cost of switching. The optimum action (harvest) maximizes

the sum of these components. Thus, the present value of the fishery equals

the expected stream of profits (payoffs) derived after adopting the optimal

harvest policy and allowing for regime switching in the stock dynamics.

The optimal value function for the problem must satisfy the following

complementarity conditions

δ V (x,R) ≥ (p− c(x)) ymax × I + µR(x, y)Vx

+1

2σ2R(x)Vxx +

∑j

λR,j [V (x, j)− V (x,R)−MR,j](4)

and

V (x,R) ≥ V (x, j)−MR,j, ∀R 6= j (5)

12

where I is an indicator taking value equal to 1, I = 1, when the industry

is open under state s, R = {s, open}, and I = 0 otherwise. The left-hand-

side of (4) is the current value of the fishery in regime R, the opportunity

cost of regime R, while the right-hand-side denotes the expected change in

the value of the fishery given by the instantaneous rent, (p − c(x)) ymax,

plus the value gained from changes in the stock, µR Vx + 1/2σ2R Vxx, in-

cluding the expected impact of a regime shift on the value of the resource,∑j λR,j [V (x, j)− V (x,R)−MR,j].

6 The second condition (5) says that the

value of remaining in regime R must be greater or equal than the value that

can be obtained after the switch to regime j minus the cost of switching.

Notice that these conditions apply to each regime and at least one of these

conditions must hold with equality for each x and R.

An alternative solution method would be to solve the model using (4)

for each regime and to use the boundary conditions given by the “value

matching” (5) and “smooth pasting” conditions at the switching points (e.g.,

figure 3) as described in Dixit and Pindyck (1994, pp.130–132) and Fackler

and Sengupta (2005). At each threshold, x∗s or x∗∗s , such conditions hold

with equality meaning that at those points the agent is indifferent between

switching and not switching regime. However, with this approach it is more

difficult to determine the optimal conditions when differential equations are

not linear in the stock variable as in this model.

6To be consistent with the theory on real option, the left-hand side of (4) can be inter-preted as the opportunity cost of holding the option to switch regime, which in equilibriummust equal the expected capital gain. The latter is made of the expected change in thevalue conditional on the current regime plus the change in value due to a regime shiftweighted by the probability of a shift (Guo et al., 2005).

13

4. An Illustration: The case of the Peruvian Anchoveta (Engraulis

ringens)

In this section the model is illustrated using data on the Peruvian anchoveta

population located along the coast of northern and central Peru (between

4 and 14°S) where the anchoveta population is more heavily concentrated

(Pauly and Tsukayama, 1987). The Peruvian anchoveta fishery started in the

early 1950s and it is the largest single-species fishery in the world (Tsukayama

and Palomares, 1987; Glantz, 1990). Landings are almost entirely processed

into fish meal destined to feed livestock, and only recently some is processed

for human consumption (Glantz, 1990; Freon et al., 2008).

The anchoveta population depends on the cold and plankton-rich up-

welling waters of the Humboldt Current (HC) and it is limited in the periods

when the seasonal upwelling is interrupted by the El Nino phenomena (e.g.,

Brainard and McLain, 1987; Chavez et al., 2003). As a result, the anchovy

population is subject to considerable interannual (years) or even multidecadal

variability, with swings in synchrony with sardine shocks throughout the

world, but especially in the area of the HC ecosystem suggesting the pres-

ence of alternating anchovy ans sardine regimes (Lluch-Belda et al., 1989;

Chavez et al., 2003; Alheit and Niquen, 2004). Moreover, there is some

indication of different stock-recruitment relationships under favorable or un-

favorable climatic conditions leading to conclude that anchovy’s recruitment

is affected by climatic anomalies in a nonlinear fashion (Cahuin et al., 2009).

Given the amount of discussion on the existence of different regimes in the

anchoveta population an application to a regime switching model to this

particular fishery seems particularly appropriate.

14

As in the theoretical model, the dynamics of the anchoveta stock is

modeled by a generalized Brownian motion where both drift and volatil-

ity terms depend on the stock level as well as on the current state, dx =

µs(x, y) dt + σs(x) dW , where s = 1, 2. Recall that the dynamic is struc-

turally different under the two states meaning that the parameters take dif-

ferent values in the two states. The growth of the stock is represented by

a logistic growth function and the stock is exploited by a fishing industry,

that is, µs(x, y) = rs(1− x/Ks)− y, where x denotes the total biomass and

y is harvest.7 The intrinsic growth rate r and the carrying capacity K of the

stock are allowed to be different under each state. The volatility of the stock

dynamics is assumed to grow with the stock and to be different in each state.

This can be interpreted as random disturbances influencing the population

and whose influence is linearly dependent to the stock level. Therefore, the

dynamics of the natural population take the following form

dx = [rsx(1− x/Ks)− y] dt+ σs x dW. (6)

The stock dynamics of the Peruvian Anchoveta is estimated using data

from Pauly et al. (1987) on estimated monthly biomass obtained from virtual

population analysis (VPA) and catch for the period 1953–1981. The catch

data are given by the total withdrawals of anchoveta that are obtained by

multiplying nominal catch by a factor of 1.2 adding consumption by guano

7The logistic growth function is standard choice for pelagic fisheries and and it is usedfor the Peruvian anchoveta as well (e.g., Freon et al., 2008). An age-structured modelwould provide a more realistic representation of this population, but such a detailed modelwould unnecessarily complicate the analysis.

15

birds, bonitos and seals to account for unreported catches and natural pre-

dation (Pauly et al., 1987).

Since the data are discrete, the continuous-time diffusion process (6) is

approximated by

xt − xt−1 + yt−1xt−1

= rs +rsKs

xt−1 + εt (7)

where ε is normally distributed with mean zero and variance σ2s . This equa-

tion is estimated allowing for two potential states in the parameters of the

stock dynamics using the MS Regress MATLAB® package provided by Perlin

(2010).8 The results show that almost all coefficients are strongly statisti-

cally significant and all have the expected sign (table ??). The estimates

provide empirical evidence for the existence of two alternative states for the

stock dynamic. The intrinsic growth rate is almost nine times larger under

state 1 indicating that state 1 has larger productivity than state 2. The

parameter indicating the carrying capacity of the stock can be derived us-

ing the estimated ratio between the intrinsic growth rate and the carrying

capacity, rs/Ks, and the estimated growth rate rs. The estimated carrying

capacity for the high state is around 25,900 thousand tons while for the low

state is around 17,400 thousand tons. The states’ variances are also highly

significant. Given these results, state 1 is characterized by higher growth

and carrying capacity and larger variability than the alternative state. Thus,

8The series are not seasonally adjusted. It is documented in the literature that seasonaladjustment distorts the dating of the turning points (shifts) because it removes informationrelevant to the regime switching (e.g., Franses and Paap, 1999; Luginbuhl and de Vos, 2003;Matas-Mir et al., 2008). Hence using seasonally unadjusted data produces more reliableestimates of the regimes.

16

state 1 is denoted as the high productivity state (H) and state 2 the low

productivity (L) state.

The estimate regime switching model predicts the Peruvian anchoveta

stock very well (figure 4). The probability of remaining in either regimes

are small indicating that both states have very small resilience and resource

switches often state. Looking at the estimated (smoothed) probability for the

highly productive state, it appears that the low productive state dominated

throughout the sample period while the high productive state was in place

mainly in the period 1959–1972 and partly in 1980–1981.

This results is supported by the existing scientific literature. There is

large evidence supporting the existence of an anchovy favorable period in

the interval 1950–1970 in the HCS due to the occurrence of a lasting pe-

riod of low temperature anomaly (e.g., Chavez et al., 2003; Cahuin et al.,

2009). Colder water temperature favours the productivity of the habitat

with positive effects on the recruitment of the anchoveta population through

density-dependent regulation. According to the estimates, the highly pro-

ductive state was working during 1959–1972, the period of the build-up of

the fishery that ended with the major collapse due to the 1972–1973 El Nino

event (Alheit and Niquen, 2004). That corresponds to a period of excep-

tional high and variable recruitment (see Pauly et al., 1987, figure 6, p.152).

Cahuin et al. (2009) also observe high recruitment rates in the 1963–1971

period, and several climatic indexes indicate a shift in the 1970s (Chavez

et al., 2003). Hence, the evidence seems to support the thesis that a highly

productive state in that period reflects the period of exceptional recruitment

of the stock due to a period of favourable climatic condition, i.e., colder

17

temperature.

5. Numerical Analysis

The optimal conditions (4)–(5) are used to numerically approximate the

thresholds for the stock that trigger the regime shift using a Matlab proce-

dure described in Fackler and Sengupta (2004). The parameter values used

for the approximation are reported in table 2. The parameters for the carry-

ing capacity are rescaled based on the capacity of state H. This means that

the solution for the optimal thresholds will be relative to it. The maximum

harvest rate, ymax, is set at the maximum harvest relative to the carrying ca-

pacity in state H, i.e., ymax = max (y/KH) observed over the sample period.

The maximum harvest rate is large compared to the maximum sustainable

yield (MSY) for the harvest, MSY= rK/4, for the high state, MSYH = 0.047,

and low state, MSYL = 0.003. Hence, fishing at maximum capacity can eas-

ily drive the stock to extinction. The values for price, cost per unit of effort,

fixed costs, and the discount rate are chosen arbitrarily.

The harvest cost function is assumed to take the following form c(x) =

c/x. The solutions need to be approximated over a carefully specified range

of values for the stock. The long-run distribution of the anchoveta stock with

no harvesting and under both states of the stock dynamics is derived to check

the range of values that the stock can take. The distribution is obtained using

the parameter values and assuming no harvest, ymax = 0. Figure 5 shows the

densities for each states. As expected, the distributions are centered around

their respective carrying capacity. According to the long-run distributions,

the interval [0, 2] covers all possible stock realizations. Thus, the state space

18

for the stock of Peruvian anchoveta used in finding the solutions is defined

by 1000 points in the [0, 2] interval.

Given the parameters of the model the optimal thresholds for the manage-

ment of the Peruvian anchoveta fishery are the following. When the resource

is in its low productivity state it is optimal to open the industry and to har-

vest at maximum capacity once the stock level crosses the level of 0.4802 from

below, x∗L. This is relative to KH . Note that this is higher than the carrying

capacity of the low state.9 Instead, it is optimal to shut-down the fishery

once the 0.2997 stock level, x∗∗L , is crossed from above. The thresholds for

the optimal harvest policy under the high productivity state are both lower

than those under the less productive state. The opening threshold, x∗H , is at

0.4722 and the closing, x∗∗H , at 0.2957. This looks reasonable given that since

the stock is more productive it can sustain some larger pressure. That is,

it is optimal to anticipate the opening and delay the closing of the industry

than in the case when the stock is less productive because it can rebound

more rapidly.

The thresholds determining the optimal harvest strategy for the Peruvian

anchoveta depend on the parameter of the model. To evaluate qualitatively

the impact of the model’s parameters on the harvest policy a number of

simulation results are performed. Increasing the probability to switch from

the low to the high state means increasing the likelihood that in the next

period the stock will be more productive than in the current period. As

9It is possible that the carrying capacity is less than one of the switch points. Forinstance, this could occur if the cost of opening the fishery is high then it would beoptimal to wait until the stock level is very high.

19

a consequence, given the fact that the switching dynamic is an exogenous

event, we may expect that the agent will increase the exploitation of the

resource even while being in the less productive state. Figure 6 shows the

opening and closing thresholds for both states. The stars indicate the optimal

solutions derived for the parameter of table 2. Both thresholds for the low

regime show a slight decrease as a result of the increasing expectation of high

natural productivity. This means that it would be optimal to anticipate the

opening of the industry, i.e., start fishing at lower population levels, and to

delay the closing, by driving the stock level to lower levels. The opposite

should apply when the stock is currently experiencing high productivity and

there is an increasing likelihood of switching to a less productive dynamic.

Figure 7 shows that the closing threshold for the high state increases until

the probability of switching to a low regime gets to 0.1 and the remains stable

thereafter. This indicates that when the expectation of a low state is very

low, i.e., less than 10 percent, it would be optimal to anticipate the closing

of the stock to exploitation.

Figures 8 and 9 illustrate the impact of varying the volatility of the

stock under the two states on the opening and closing thresholds. Increas-

ing volatility means increasing the uncertainty of the stock dynamics. This

simulations show that when the dynamic of stock becomes more noisy and

hence more uncertain the difference between the optimal thresholds of the

low productive regime increases. Specifically, the opening threshold increases

with uncertainty, while the closing threshold decreases. This means that it

is optimal to delay the opening (closing) of the industry because the op-

tion of waiting to open (close) has more value when uncertainty is higher.

20

The thresholds for the high productive regime appears to be insensitive to

the uncertainty of a low productive resource. The difference between the

optimal thresholds of the high productive regime also increases in a similar

fashion. Thus it appears that under both states of the dynamics it is optimal

to maintain the current status of the industry until uncertainty resolves than

paying the switching costs, as pointed out for the single state case analyzed

by Nøstbakken (2006). Interestingly, the hysteresis due to the presence of

alternative dynamics becomes larger with uncertainty.

The simulations show that when industry capacity increases it is optimal

to start fishing at smaller stock levels and stop fishing at higher stock levels

(figure 10). This indicates that the option of waiting to open (close) be-

comes less valuable as the capacity of the industry increases. The maximum

capacity indicates the maximum harvest rate of the fishing industry. The

capacity can be increased by allowing more fishermen into the industry or

allowing existing fishermen to increase their input levels. The hysteresis due

to the alternative states of the stock dynamic decreases as the capacity in-

creases.10 The thresholds tend to converge meaning that the optimal policy

would not be different for different stock dynamics.11 Therefore, if the ca-

pacity of the industry increases, the management agency should intervene by

reducing/increasing the stock levels at which the stock can start/stop being

exploited.

As expected, increasing the fixed opening costs delays the opening to

10Specific plots can be provided.11These results are consistent with Nøstbakken (2006) where, at low output price,

switching curves on the price–stock space move outwards as capacity increases and theydo so in a less than proportional fashion.

21

fishing (figure 11). In this situation, it is more profitable for the agent to

maintain the status of the industry for a while until the situation improves,

then the fishery can be opened. The closing thresholds decreases as a result

of higher opening costs. This indirect effect are similar to what observed

by Mason (2001) for mines. When opening becomes more expensive, the

agency that closes the industry will be less willing to reopen, which would

reduce the value of the option to open. Also, the increase of closing costs

delays the closing of the fishery and would induce the agency to become more

conservative in opening the mine (figure 12). Interestingly, the hysteresis

becomes larger at the fixed costs increase. The gap induced by the alternative

resource states is small at low start-up/closing costs and it gets reversed

(thresholds cross) and increases for higher fixed costs.

6. Forthcoming Analysis

In this section I extend the model to the case of endogenous switching. En-

dogenous switching means that the transitional probabilities are set to be

explicit functions of the stock itself plus some exogenous factors that also

influence the probability of shifting the state of the stock dynamics, i.e.,

λR,j(x, z), where z are the exogenous factors.

One implication for restricting to exogenous switching is that the agency

only reacts to the new biological conditions. Instead, allowing for endogenous

regime switching the agency may influence the likelihood of changing state

by adjusting its management policy. This raises a richer set of implications

for the management policy of the resource.

[...]

22

7. Conclusions

[...]

23

Table 1: MS regression model

Variable Coefficient (Std. Err.)r1 0.1896 (0.0161) ***r2 0.0210 (0.0089) **r1/k1 -7.33e-6 (1.38e-06) ***r2/k2 -1.21e-6 (1.14e-06)

σ21 0.0035 (0.0009) ***σ22 0.0026 (0.0005) ***

λ1 0.13 (0.02) **1− λ1 0.87 (0.08) ***λ2 0.08 (0.03) **1− λ2 0.92 (0.08) ***

log likelihood 286.268N.obs 347

Note: Double asterisks (**), and triple asterisks (***) denote significance at 5%, and 1% levels.

Table 2: Parameter values

Param. Value DescriptionrH 0.1896 Growth rate HighrL 0.0210 Growth rate LowKH 1 Carrying capacity HighKL 0.67 Carrying capacity LowσH 0.0588 Stock volatilityσL 0.0510 Stock volatilityλH 0.08 Prob. to shift to HighλL 0.13 Prob. to shift to Lowymax 0.1108 Industry capacityp 1 Pricec 0.3 Cost per unit of effortMo 0.01 Cost of openingMc 0.01 Cost of closingρ 0.1 Discount rate

24

-•

0 x∗∗s

Closed Inaction

x∗s

Open

K

Figure 1: Thresholds under state s (K indicates the maximum attainable stock)

-•

0 x∗∗H x∗∗L x∗H x∗L K

Figure 2: Thresholds (K indicates the maximum attainable stock)

-•

0

x∗∗H

0.2957

x∗∗L

0.2997

x∗H

0.4722

x∗L

0.4802 K

Figure 3: Thresholds (K indicates the maximum attainable stock)

25

Figure 4: Peruvian Anchoveta

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

5000

10000

15000

20000

25000

Switc

hing

 Proba

bility

Thou

sand

 Ton

s

Biomass Predicted Biomass Probability High Productive State

Figure 5: Long-run stock density

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Stock level, X

Den

sity

High state

Low state

26

Figure 6: The impact of λH on the opening thresholds

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.25

0.3

0.35

0.4

0.45

0.5

Switching probability, low to high

Sto

ck le

vel,

X

*

*

*

*

Low−StateHigh−State

Figure 7: The impact of λL on the closing thresholds

0 0.05 0.1 0.15 0.2 0.250.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Switching probability, high to low

Sto

ck le

vel,

X

*

*

*

*

Low−StateHigh−State

27

Figure 8: The impact of σL on the optimal thresholds

0 0.05 0.1 0.15 0.2 0.250.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

Volatility low state

Sto

ck le

vel,

X

*

*

*

*

Low−StateHigh−State

Figure 9: The impact of σH on the optimal thresholds

0 0.05 0.1 0.15 0.2 0.250.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

Volatility high state

Sto

ck le

vel,

X

*

*

*

*

Low−StateHigh−State

28

Figure 10: The impact of the maximum capacity on the optimal thresholds

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

Industry capacity

Sto

ck le

vel,

X

*

*

*

*

Low−StateHigh−State

29

Figure 11: The impact of the opening fixed cost on the opening thresholds

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Opening fixed cost

Sto

ck le

vel,

X

****

Low−StateHigh−State

Figure 12: The impact of the closing fixed cost on the closing thresholds

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Closing fixed cost

Sto

ck le

vel,

X

****

Low−StateHigh−State

30

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